Einstein simultaneity: just a convention?

[DrGreg]

Right, so although it's common to state that the Minkowski norm is "coordinate independent", that's only true within a coordinate subclass. What we need to know is, what is the core principle that unites the Minkoswki norm with the radio norm? A mathematician could probably say it in one line, but I wouldn't understand a single word-- I want the physical statement, and I feel that we should teach relativity to reflect that, rather than asserting a constant speed of light as if it were a physical fact (that is very much what is normally done).

The interval "ds²" is invariant -- the same value between a given pair of nearby events according to every observer. If the observer is using standard Einstein-synced Minkowski coords, the interval is always given by the formula

$$ds^2 = dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2$$​

(the c might be in a different place or the signs might be opposite according to what your metric sign convention is, but once that's decided, all observers use the same formula).

The physical significance of the interval (as I chose to write it) is:

- if ds² > 0, ds is the proper time taken by an inertial observer to travel between the events (and it is also the longest proper time that anyone, inertial or not, could take to travel between the events) "Proper time" means time measured by your own clock between events that occur at zero distance from yourself (so no sync required).

- if ds² < 0, $\sqrt{-c^2ds^2}$ is the proper distance between the events measured by an inertial observer who considers them to be Einstein-simultaneous

- if ds² = 0, it is possible for a photon of light to pass through both events.

If you use coordinates other than the standard orthogonal Einstein-synced coords, you will get a different formula for ds².

For example, even with Einstein-synced time but spherical polar spatial coords, you get

$$ds^2 = dt^2 - dr^2 / c^2 - r^2 ( d\theta^2 + \sin^2 \theta d\phi^2) / c^2$$​

In Special Relativity (SR), you never use coordinates like this, but in General Relativity (GR), you have no choice but to do so. Special Relativists almost always use Einstein-synced orthogonal Minkowski coords, but General Relativists are happy to use any coordinate system you like. (But the maths of GR is a whole lot more complicated than SR.)

On a final note, I believe the second postulate should really be interpreted as "the motion of a photon is independent of whatever emitted it", so that it is impossible for one photon to overtake another traveling in the same direction. The fact that all inertial observers using Einstein-synced clocks agree on the value of the coordinate speed of light is then really a consequence of the first postulate (because otherwise you could distinguish one frame from another). (See "Two myths about special relativity", Ralph Baierlein, http://link.aip.org/link/?AJPIAS/74/193/1 , section III.)

 

[shoehorn]

I disagree strongly with this statement. The only philosophical aspect of special relativity of which I am aware is Occham's Razor

Really? What about the substantivalism/relationism debate? Spacetime pointillisme? The relation of simultaneity to the automorphism group of Minkowski space?

Those are three massive philosophical questions which arise out special relativity. The substantivalist/relationist debate in particular is (or at least should be) encountered by most everyone who studies the philosophy of science at university.

 

[Ken G]

The interval "ds²" is invariant -- the same value between a given pair of nearby events according to every observer. If the observer is using standard Einstein-synced Minkowski coords, the interval is always given by the formula

$$ds^2 = dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2$$​

Yes, that is the standard coordinatization, and I realize what all that gives rise to. My point is that this is usually the starting point for relativity, so there is no recognition that any other choice is even possible. I want to understand what must be true before we make this choice, what have we learned about reality, not about what a good coordinate is.

If you use coordinates other than the standard orthogonal Einstein-synced coords, you will get a different formula for ds².

Right, so the question is, what really is coordinate independent?

On a final note, I believe the second postulate should really be interpreted as "the motion of a photon is independent of whatever emitted it", so that it is impossible for one photon to overtake another traveling in the same direction.

But that is a much weaker version, because different sources could still emit photons with the same properties but not isotropic speeds. And if the postulate is "space plays no special role in the propagation, it's always the same", then note this only applies if we treat inertial observers as special. In short, the "special" in special relativity is somewhat oxymoronic.

 

[Hurkyl]

What we need to know is, what is the core principle that unites the Minkoswki norm with the radio norm?

Right, so the question is, what really is coordinate independent?

I'll try anyways -- the answer is "lengths and angles".

Of course, the more precise version of that answer includes:
The spacelike / lightlike / timelike classification of (tangent) vectors
Proper length of spacelike paths
Proper duration of timelike paths
Circular angle in a spatial plane
Hyperbolic angle in a mixed temporal/spatial plane

There are other coordinate-independent things too, of course, such as the topology of space-time, or the mere fact that global Einstein synchronization is possible.

 

[Ken G]

The spacelike / lightlike / timelike classification of (tangent) vectors

This one seems pretty important, with its connection to causality.

Proper length of spacelike paths

I'm not so sure about this one, spacelike paths sound like pure conceptualization to me so might well be coordinate dependent, or even not invoked at all.

Proper duration of timelike paths

This is clearly a key invariant, as we can directly measure it. I think this is the crucial invariant around which a theory should be built, and that's the main advantage of radio coordinates.

Circular angle in a spatial plane
Hyperbolic angle in a mixed temporal/spatial plane

These two sound like they are connected to something important, the spacetime curvature that becomes so important for gravity, but by themselves they sound coordinate dependent to me. I'm not sure how you measure a circular angle, or an angle in spacetime, and adding angles in a triangle requires assumptions about the vertices, so I can't say for sure if these are dependent on how we conceptualize spacetime or not.

There are other coordinate-independent things too, of course, such as the topology of space-time, or the mere fact that global Einstein synchronization is possible.

Yes, topology must be fundamental. The fact that Einstein synchronization is possible might not be so fundamental, there might always be a way to do it for objects with more general properties than our reality. So it might not actually be saying anything about reality, more so than about simultaneity conventions. We need a complete mathematical understanding of what the possibilities are.

In any event, if some or even all of the above list are fundamental properties of any successful description of reality, we still need to express those fundamental properties in the most general way, and yet also the way that incorporates everything that the observations show. In short we don't want to imagine we need to assume anything, just to make our life simpler, that is not required to fit the observations, nor do we want to leave anything out that observations require we include. I do not see at the moment why the standard postulates of relativity accomplish that, so that's more or less what I'm asking.

 

[Hurkyl]

I'm not so sure about this one, spacelike paths sound like pure conceptualization to me so might well be coordinate dependent, or even not invoked at all.

I agree it's a little harder to imagine, but I do think it's still important. For example, it's lurking behind the scenes when we talk about (coordinate) length -- if we choose an inertial coordinate chart and ask for the length of a piece of string relative to that chart, we are actually asking for the proper length of the spacelike path defined by the string and a hyperplane of simultaneity. Now, if we choose to work with different coordinates, if we are still able to identify that spacelike path, we can compute its proper length and get the same answer as before. In fact, I think it's a very good exercise to derive the length contraction formula using just this idea. (And it might help with understanding the barn-and-pole pseudoparadox)

Another point is that, over 'infinitessimal' distances, each observer has a (spacelike) hyperplane of simultaneity, which can be useful for defining spacelike paths.

These two sound like they are connected to something important,

The 'circular' angle is just ordinary Euclidean angles. ('circular' because angles are based on the circle, and specifies which trigonometry is appropriate)

The angle in a mixed spatial-temporal plane corresponds to relative velocity -- something that is presumably physically observable when two objects pass by each other. (Sorry, my brain was firmly in 'geometry' mode) Angle measure in such a plane is based on the hyperbola, and uses the hyperbolic trig functions. I think 'rapidity' is the term physicists use instead of 'angle'.

The fact that Einstein synchronization is possible might not be so fundamental, there might always be a way to do it for objects with more general properties than our reality.

Nonetheless, asserting it's possibility is still a very strong assertion, and one which reality is known to violate over non-'infinitessimal' length and time scales. (I suspect it's very nearly equivalent to the special relativistic requirements on space-time, but I haven't tried working out the detail)

 

[Ken G]

Now, if we choose to work with different coordinates, if we are still able to identify that spacelike path, we can compute its proper length and get the same answer as before. In fact, I think it's a very good exercise to derive the length contraction formula using just this idea.

But length contraction is part of my issue with proper length. When we change inertial frames (by changing our velocity with respect to the rod), we infer a new length when we intercept the simultaneity hyperplane with the rod as you mention. When we back that out into the length in the frame of the rod, it returns to the correct length. But if we accelerate the rod, we have to achieve the same shrinking manually, with varying proper accelerations, to keep the rod the same length in its own frame. So that time we did do something physical to achieve the length contraction. This feels rigged to me, I sense too much of our own fingerprints "at the crime scene".

Another point is that, over 'infinitessimal' distances, each observer has a (spacelike) hyperplane of simultaneity, which can be useful for defining spacelike paths.

Yes, I've wondered about this too, it doesn't seem like we can do away with spacelike separation completely, we seem to need it in a kind of tangent space we carry around with us (rulers and whatnot). But integrating that to get finite proper distances is what doesn't make a lot of sense to me, physically that seems like something arbitrary, unlike integrating proper time, which shows on a clock.

The 'circular' angle is just ordinary Euclidean angles. ('circular' because angles are based on the circle, and specifies which trigonometry is appropriate)

But how do we measure it? You can't really measure the angle around a point, because you don't know if your protractor is warped. And if you add the angles in a triangle, you need to define the locations of the vertices. You'll probably pick a single inertial frame, but why-- why does a triangle in such a frame define something about angles? Are we learning something fundamental about reality, or just something about our biases toward frames with no forces in them? General relativity tells us that gravity can mess with those angles even in an inertial system, but choosing inertial systems is still just choosing a coordinatization, it seems to me. It seems like there's something underneath that which is more fundamental than a bias toward conceptualizing spacetime using inertial frames.

The angle in a mixed spatial-temporal plane corresponds to relative velocity -- something that is presumably physically observable when two objects pass by each other.

Relative velocity is not directly measurable, but redshift/blueshift is, so we can certainly pin our theory on a need to get the right answer for that. Certainly when there is gravity around, we need a more general concept than relative velocity, but I realize we are talking special relativity here. So is there really a physical concept of relative velocity even without any gravity? I'm not so sure that's a physical concept, it seems like yet another coordinate choice.

Nonetheless, asserting it's possibility is still a very strong assertion, and one which reality is known to violate over non-'infinitessimal' length and time scales. (I suspect it's very nearly equivalent to the special relativistic requirements on space-time, but I haven't tried working out the detail)

I agree with that in the way special relativity is normally constructed, and the fact that it works (in the absence of gravity) definitely restricts the universe in some important way. But I think what we tend to do is to effectively invert the Einstein prescription into a picture of how the universe works. If the mapping from all the universes that admit that description is not one-to-one onto the observation set we have at our disposal, that inverse mapping does not necessarily describe the universe correctly.

That's what I'm asking for here-- a set of postulates that not only correctly describe all the observations, but are also the minimal set that do so, so can be inverted into the full set of possible universes we are constraining. One must be cautious about inverting projections. Take the idea that all inertial observers are created equal in "the eyes of the law", if you will. That is usually framed as a fundamental statement about reality, but when one realizes that "inertial observer" just means "observer who can account for everything that is happening in terms of forces on observed objects", should we be surprised that such observers can indeed account for everything using one unified prescription? Haven't we simply excluded the observers who are going "what the heck...?"

 

[phyti]

I disagree strongly with this statement. The only philosophical aspect of special relativity of which I am aware is Occham's Razor, or, as Einstein said, "Everything should be made as simple as possible, but not simpler."

Relativity was not developed and accepted because of some philosophical crusade in the scientific community at the time. It was developed and accepted on the exact same basis as all other successful scientific theories: it was the simplest theory that fit the observed experimental data. Classical physics couldn't explain the data, and other theories that could explain the data (like Lorentz's ether) were more complicated. That is pure science, and other than Occham's Razor I really see very little philosophical in it.

The Michelson-Morley 1887 experiment supported the constant 'measured' speed of light. The body of scientific knowledge in 1900 was, by today's standard, very limited in scope and area of application. The rules of physics were derived from experiments confined to Earth (except astronomical observations), and there was never a concerted effort to prove their universality. It was a gigantic extrapolation to state "the rules are universal for all inertial frames".
Einstein preferred a deterministic behavior of the world, a common view then. This is emphasized by his objection to the randomness of quantum theory. Human nature likes a secure world with no surprises or strange behavior.
This is why I say the 1st postulate was a philosophical preference.
If the speed of light is "constant and independent of its source", then it should be possible to derive the effects of uniform motion on measurements using this postulate alone.

 

[Antenna Guy]

Yes, I've wondered about this too, it doesn't seem like we can do away with spacelike separation completely, we seem to need it in a kind of tangent space we carry around with us (rulers and whatnot). But integrating that to get finite proper distances is what doesn't make a lot of sense to me, physically that seems like something arbitrary, unlike integrating proper time, which shows on a clock.

Consider looking into what a Fresnel (pronounced fra-nel) region is.

Regards,

Bill

 

[DrGreg]

The angle in a mixed spatial-temporal plane corresponds to relative velocity -- something that is presumably physically observable when two objects pass by each other. (Sorry, my brain was firmly in 'geometry' mode) Angle measure in such a plane is based on the hyperbola, and uses the hyperbolic trig functions. I think 'rapidity' is the term physicists use instead of 'angle'.

Relative velocity is not directly measurable, but redshift/blueshift is, so we can certainly pin our theory on a need to get the right answer for that. Certainly when there is gravity around, we need a more general concept than relative velocity, but I realize we are talking special relativity here. So is there really a physical concept of relative velocity even without any gravity? I'm not so sure that's a physical concept, it seems like yet another coordinate choice.

Yes, the "hyperbolic angle" between two timelike vectors is called "rapidity" and it is equal to $\log_e k$, where k is the doppler factor (emitted frequency)/(observed frequency), which can be measured using only proper time. (The two timelike vectors are the 4-velocities of emitter and observer.) Note that if you rescale rapidity to be $c \log_e k$ then it approximates to coordinate-speed at low speeds.

In terms of general relativity, it only makes unambiguous sense to measure rapidity "locally" i.e. for two observers passing by each other, so that gravitational doppler shift is excluded from consideration. In G.R. only local measurements have physical significance; "remote" measurements get distorted by the curvature of spacetime and tend to be dependent on non-physical coordinates.


"Proper distance" between two objects that are stationary relative to the observer requires no definition of simultaneity as you can take as long as you like to compare your objects against a ruler. It's only the measurement of moving objects that requires a clock synchronisation convention. The distance between two events is the proper distance between two stationary objects each of which experiences one of the events.

So the interval ds can be defined in terms of proper time (if timelike) or proper distance (if spacelike), neither requiring clock synchronisation.

Note that if you have a definition of "spatial distance" for stationary objects then you can define spatial angle via the Cosine Rule $dc^2 = da^2 + db^2 - 2 \, da \, db \, \cos C$.

On a final note, I believe the second postulate should really be interpreted as "the motion of a photon is independent of whatever emitted it", so that it is impossible for one photon to overtake another traveling in the same direction.

But that is a much weaker version, because different sources could still emit photons with the same properties but not isotropic speeds.

Yes, as I stated it, my 2nd postulate is weaker than the common interpretation, because I demand no coordinate system. "Isotropic speed" implies a coordinate system to measure speed.

And if the postulate is "space plays no special role in the propagation, it's always the same", then note this only applies if we treat inertial observers as special. In short, the "special" in special relativity is somewhat oxymoronic.

"Special" means "ignoring gravity", rather than the status of inertial observers.

Inertial observers are different to all other observers, in a physically measurable way: they do not experience proper acceleration, i.e. "G-forces", something they can determine using an appropriate accelerometer device, without a coordinate system. (And this definition works in GR as well as SR. Inertial observers still have special status in GR, but they no longer travel at constant velocity relative to each other.)



Forgive me if I'm explaining something you already know, Ken, but the mathematical description of spacetime makes a distinction between a 4D vector X and its components (t,x,y,z). You can switch between lots of different coordinate representations, but they all represent the same vector which exists independently of its coordinates. Spacetime is equipped with an scalar "inner product" g(X,Y) which is analogous to the "dot product" of 3D Euclidean vectors x.y. The inner product, or "metric" is invariant, that is you always get the same answer for g(X,Y) no matter what coordinate system you use to carry out the calculation. The properties of spacetime can described in terms of the properties of the metric (e.g. g(X, Y+Z) = g(X,Y) + g(X,Z) etc etc). And then ds² = g(dX,dX).

So, mathematically, spacetime is defined as a four dimensional vector space equipped with a metric g that satisfies certain conditions (which can all be expressed in a coordinate-free vector notation). I provide this as background information, as I know you are really looking for a physical rather than mathematical model.

 

[Ken G]

Note that if you rescale rapidity to be then it approximates to coordinate-speed at low speeds.

True, but that's not really a speed, it's a Doppler factor. That's the thing we can measure, speed requires a coordinatization.

In terms of general relativity, it only makes unambiguous sense to measure rapidity "locally" i.e. for two observers passing by each other, so that gravitational doppler shift is excluded from consideration. In G.R. only local measurements have physical significance; "remote" measurements get distorted by the curvature of spacetime and tend to be dependent on non-physical coordinates.

Indeed, and my question is, are we really sure this is a gravitational effect? Maybe that is just as true in special relativity, only we have chosen to pretend otherwise because there exists a transparent globalization (based on inertial frames) in the absence of gravity.

"Proper distance" between two objects that are stationary relative to the observer requires no definition of simultaneity as you can take as long as you like to compare your objects against a ruler. It's only the measurement of moving objects that requires a clock synchronisation convention. The distance between two events is the proper distance between two stationary objects each of which experiences one of the events.

That had me thinking for awhile, but I don't think that would give a unique result. After all, there are infinitely many pairs of mutually stationary objects that could have one object at each event, all with different distances between them. If you further stipulate that the objects must be stationary with respect to the observer doing the measurement, it just means each such pair comes with their own observer, each finding a different "proper distance" between the events. If the events themselves don't have a concept of being "stationary", which they don't normally, then we still have no way to know which observer is getting the "proper" result.

That's the problem with using objects to witness events, that's really something observers should be doing, and using pairs of observers, instead of a single observer, seems to introduce ambiguities. That's why I never understood the concept of proper distance, and still don't. It seems purely coordinate dependent.

Yes, as I stated it, my 2nd postulate is weaker than the common interpretation, because I demand no coordinate system. "Isotropic speed" implies a coordinate system to measure speed.

I agree-- if we substitute your way of stating the second postulate, it would be interesting to see what possibilities would still be considered admissible ways of looking at reality. Ironically, that way of stating the postulate is normally associated with the presence of a wave medium, not the absence of one.

"Special" means "ignoring gravity", rather than the status of inertial observers.

I don't agree there, to me "special", as it is normally used, means "elevate the importance of inertial observers" (in things like simultaneity conventions, etc.) as being the ones for whom the laws of physics are the same. That seems prejudicial toward Galileo's principle of inertia, which is a kind of circular reasoning-- if we make one of the laws be that there is no acceleration without forces, then of course we are going to think the laws have a special relationship with acceleration-free frames. In "general" relativity, all observers, even the accelerated ones, are on an equal footing, because we treat "real" and "coordinate" forces in a unified way (gravity itself being hard to categorize as one or the other).

Inertial observers are different to all other observers, in a physically measurable way: they do not experience proper acceleration, i.e. "G-forces", something they can determine using an appropriate accelerometer device, without a coordinate system.

They are not "different"-- everyone can measure something with an accelerometer. The inertial ones are simply defined as those who measure zero.

Forgive me if I'm explaining something you already know, Ken, but the mathematical description of spacetime makes a distinction between a 4D vector X and its components (t,x,y,z).

Yes, that's an important issue, how we obtain those components.

You can switch between lots of different coordinate representations, but they all represent the same vector which exists independently of its coordinates. Spacetime is equipped with an scalar "inner product" g(X,Y) which is analogous to the "dot product" of 3D Euclidean vectors x.y. The inner product, or "metric" is invariant, that is you always get the same answer for g(X,Y) no matter what coordinate system you use to carry out the calculation.

Not if you use "radio coordinates". This is part of the point-- the metric space has more general properties than the form of the metric.

So, mathematically, spacetime is defined as a four dimensional vector space equipped with a metric g that satisfies certain conditions (which can all be expressed in a coordinate-free vector notation).

Right-- but that by itself won't get you the Minkowski norm that we teach as if it was an inherent part of the metric space. If we use different coordinates, we get a different form for the metric, but the physics is identical. So what's the real physics here?

 

[Dale]

I don't agree there, to me "special", as it is normally used, means "elevate the importance of inertial observers" (in things like simultaneity conventions, etc.) as being the ones for whom the laws of physics are the same.

You seem to have a misunderstanding. There is the general theory of relativity, which simplifies to the special theory of relativity in regions of flat spacetime (i.e. special relativity means "special case" of the more general theory of relativity. Special relativity, in turn, simplifies to galilean relativity for v<<c.

You certainly can have accelerating observers and all sorts of forces in special relativity.

 

[Dale]

Really? What about the substantivalism/relationism debate? Spacetime pointillisme? The relation of simultaneity to the automorphism group of Minkowski space?

Those are three massive philosophical questions which arise out special relativity. The substantivalist/relationist debate in particular is (or at least should be) encountered by most everyone who studies the philosophy of science at university.

People can and will debate about anything. And although the debate may even be very important, it is not an essential part of the theory itself. As a case in point I have used SR for years (not professionally) and I have no idea what you are talking about with any of those debates. It isn't that the debates are unimportant, they are just not essential to the theory.

 

[Hurkyl]

Yes, that's an important issue, how we obtain those components.Not if you use "radio coordinates". This is part of the point-- the metric space has more general properties than the form of the metric.
Right-- but that by itself won't get you the Minkowski norm that we teach as if it was an inherent part of the metric space. If we use different coordinates, we get a different form for the metric, but the physics is identical. So what's the real physics here?

He didn't say "the coordinate expression for the metric is invariant" -- he said "the metric is invariant".

Compare -- lengths and angles are invariants of Euclidean geometry, even though formulas for computing them can have varying forms between different coordinate charts.

 

[Hurkyl]

In "general" relativity, all observers, even the accelerated ones, are on an equal footing, because we treat "real" and "coordinate" forces in a unified way (gravity itself being hard to categorize as one or the other).

In general relativity, "gravity" is simply the tendency of objects to travel in a straight line through space-time; i.e. inertial travel. An object under the sole influence of gravity travels in a straight-line path (a geodesic), and experiences a net force of zero.

(note: force is an invariant of motion. Furthermore, the notion of force is very different from the notion of coordinate acceleration)

 

[Ken G]

You seem to have a misunderstanding. There is the general theory of relativity, which simplifies to the special theory of relativity in regions of flat spacetime (i.e. special relativity means "special case" of the more general theory of relativity. Special relativity, in turn, simplifies to galilean relativity for v<<c.

You certainly can have accelerating observers and all sorts of forces in special relativity.

I am aware of that, my point was the reason for the word "special" in the title of the theory has to do with the specialness of inertial observers, not the specialness of the absence of gravity. Inertial observers are the observers for whom the postulates of special relativity apply, that's pretty special. Accelerated observers can be treated, but not directly with the postulates, one must first find an inertial frame, apply the postulates there, and convert to the accelerated frame to find the metric that applies to the accelerated frame (such as the Rindler metric for uniform acceleration). The postulates of special relativity are wrong in an accelerated frame. They are also wrong when there's gravity, so that's why gravity had to be added, but what made it "general" was the ability to treat all observers with the same formalism.

 

[Ken G]

He didn't say "the coordinate expression for the metric is invariant" -- he said "the metric is invariant".

Indeed, and he expressed that metric as an explicit function g(dX,dX). That function won't apply for, say, radio coordinates, so if that function is interpreted as "the metric" (the clear insinuation), then it is not invariant for all inertial observers in all coordinate charts. Orthonormal charts only, i.e., Lorentz transformations-- that's the problem.

Compare -- lengths and angles are invariants of Euclidean geometry, even though formulas for computing them can have varying forms between different coordinate charts.

Only if you either restrict to orthonormal coordinate charts, or redefine what you mean by "distance" to assure that it is preserved in any coordinate chart, which makes it an invariant by construction not by geometry. In other words, I do not need to define a concept of "distance" the same way the Minkowski norm does, nor do I need to agree with its answers for that quantity, to be doing the same physics-- and we should find a way to teach it that reflects that. Does the normal way?

 

[Ken G]

In general relativity, "gravity" is simply the tendency of objects to travel in a straight line through space-time; i.e. inertial travel.

Like I said, it's hard to classify as "real" or "coordinate" in nature (coordinate being in how it doesn't show up in a locally inertial coordinate system, or as you say creates travel in a straight line, until you look at tidal effects on more than one particle, and real being how it indeed exhibits tidal effects on multiple particles).

(note: force is an invariant of motion. Furthermore, the notion of force is very different from the notion of coordinate acceleration)

I believe you are distinguishing proper acceleration from coordinate acceleration. An important distinction in general, but I'm missing the specific relevance here. The reason I said it's tricky to categorize gravity is because locally it is governed by the equivalence principle, so its affect on a single free particle cannot be detected, but if you have several particles, you can detect it as a real effect even without there being the presence of any proper acceleration. Does that count as a real force, or a coordinate force?

 

[Dale]

I am aware of that, my point was the reason for the word "special" in the title of the theory has to do with the specialness of inertial observers, not the specialness of the absence of gravity.

No, it is neither. It is "specialness" of flat spacetime which is not the same thing as the absence of gravity.

In GR the worldline of an inertial observer is a geodesic, in SR the worldline of an inertial observer is a straight line. By definition, a flat spacetime is one where all the geodesics are straight lines. So SR is the special case of GR in a flat spacetime.

Inertial observers are the observers for whom the postulates of special relativity apply, that's pretty special.

You are mixing up "observers" and "reference frames". The postulates of SR apply to equally well to all observers, inertial and non-inertial, as analyzed in an inertial reference frame. You can use the postulates of SR without modification to analyze a non-inertial observer from an inertial reference frame. You cannot use the postulates of SR without modification to analyze even an inertial observer from a non-inertial reference frame.

 

[Ken G]

No, it is neither. It is "specialness" of flat spacetime which is not the same thing as the absence of gravity.

Wiki confirms my expectaton: "The theory is termed "special" because it applies the principle of relativity only to inertial frames." (http://en.wikipedia.org/wiki/Special_relativity)

You are mixing up "observers" and "reference frames".

I wouldn't say that, though your issue is largely semantic. In Einstein's approach to special relativity, which is what I am questioning, there is no difference, as he extends the inertial observer to an entire global frame, and noninertial observers are referenced to inertial ones instantaneously comoving. Of course this is not done in general relativity, but that only underscores my issue with thinking that the Einstein conventions are anything but a coordinate convenience. I don't know that I'm right, but so far there have been no successful challenges.

The postulates of SR apply to equally well to all observers, inertial and non-inertial, as analyzed in an inertial reference frame.

By invoking that frame, you are invoking an observer, to give the values in that frame scientific meaning. But it's something of a moot point, I am saying that what is "special" about it is that you cannot use its postulates to infer what a noninertial observer will measure, unless you first reference the noninertial observer to one in an inertial frame, do the calculation in that frame, and then transform back. Specifically, neither the form of the laws of physics, nor the speed of light, will be the same for any noninertial observer. In short, the postulates won't work for that observer, which is very much the spirit of special relativity.

You can use the postulates of SR without modification to analyze a non-inertial observer from an inertial reference frame.

I know that. Nevertheless, it violates the idea that "the laws of physics are the same for all observers", instead it becomes "the laws of physics become the same for all observers only once they are translated to an inertial frame". If you're going to do a translation, you might just as well pick an absolute frame and always transform to that-- it is still a violation of the spirit of relativity, and that's just what general relativity fixes via formal unification of accelerating frames with gravity using the equivalence principle.

 

[Antenna Guy]

In GR the worldline of an inertial observer is a geodesic, in SR the worldline of an inertial observer is a straight line. By definition, a flat spacetime is one where all the geodesics are straight lines. So SR is the special case of GR in a flat spacetime.

Can a geodesic in GR be a straight line?

What curves a geodesic in GR?

Regards,

Bill

 

[Dale]

Wiki confirms my expectaton: "The theory is termed "special" because it applies the principle of relativity only to inertial frames." (http://en.wikipedia.org/wiki/Special_relativity)

If you are going to try to win an argument by appeal to authority you should at least try to do better than Wikipedia. RL Faber. "Differential Geometry and Relativity Theory: An Introduction" has a whole chapter entitled "Special Relativity: the Geometry of Flat Spacetime". Or MS Parvez. "On the theory of flat spacetime" which says in the abstract "Special relativity, in essence, is a theory of four-dimensional flat spacetime".

I note that you didn't address the point I made about geodesics.

it violates the idea that "the laws of physics are the same for all observers", instead it becomes "the laws of physics become the same for all observers only once they are translated to an inertial frame".

Sorry Ken, you need to read up a little more. The first postulate is in fact closer to your second statement than the first. The first postulate is, in Einstein's words, "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good ... The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion".

 

[Dale]

Can a geodesic in GR be a straight line?

Yes, anywhere the space is not curved.

What curves a geodesic in GR?

The intrinsic curvature of the space, which is in turn caused by the presence of energy.

 

[bernhard.rothenstein]

conentionality

I'm curious about how people here view Einstein's prescription for determining simultaneity in an inertial frame, and how the extension of that approach to other inertial frames spawns the Lorentz transformation. It seems to me the competing pictures here are that this is an arbitrary way (in the sense of, not physically forced, even if convenient) to coordinatize time, and hence the Lorentz transformation is an arbitrary mapping between the coordinates of different reference frames, versus saying that the Einstein convention is fundamental to what we mean by time, and the Lorentz transformation is fundamental to what we mean by motion. I am rather of the former school, that what is physically fundamental is a deeper symmetry that allows the Einstein convention to be a particularly convenient coordinate choice, but that its physical significance comes entirely from how it simplifies the coordinatizations when we apply the laws of physics. But others might argue that the simplification is so fundamental that it would be foolish for us to imagine that "reality itself" could be doing anything different, even if just a means for recognizing equivalent possibilities.

Note, in particular, that the isotropic and constant speed of light in an inertial frame is a ramification of Einstein's coordinatization prescription, so an equivalent way to ask this is, is the isotropic speed of light a law of nature or just the proof that there exists a particularly elegant coordinate possibility? As the former is often taken as a postulate of special relativity, are we messing up the proper axiomatic structure of our art here?

Please follow the following thoughts:
1.Einstein's transformation equations hold only with Einstein synchronized clocks t(E) and t"(E).
2.the t(E) and t'(E) readings could be brought in a physically correct relationship with the readings of other clocks synchonized in a different way. In the case of the synchronized transformation (Mansouri and Sexl, Abreu and Homen, Guerra) or inertial transformations (Selleri) it reads
t(E)=t(v)+Vx/cc (1)
With (1) the Lorentz transformation becomes
t'(E)=t(v)/(1-VV/cc)
the concept of simultaneity having an absolute character.
3.It is considered that under such conditions the reference frame I is in absolure rest relative to the ether the motion of I' relative to it having an absolute character confering to I' some properties (anisotropy).
4.My oppinion is that those properties are merely introduced by the shift from the reading t(E) to the reading t(v). Using different physically correct relationships between t(E) and the reading t of a differently synchronized clock we obtain different transformation equations which confer different properties to I'.
I would highly appreciate your oppinion.

 

[Hurkyl]

Indeed, and he expressed that metric as an explicit function g(dX,dX). That function won't apply for, say, radio coordinates, so if that function is interpreted as "the metric" (the clear insinuation), then it is not invariant for all inertial observers in all coordinate charts. Orthonormal charts only, i.e., Lorentz transformations-- that's the problem.

Again, you're confusing "the metric" with "coordinate representation of the metric"

Compare -- lengths and angles are invariants of Euclidean geometry, even though formulas for computing them can have varying forms between different coordinate charts.

Only if you either restrict to orthonormal coordinate charts, or redefine what you mean by "distance"

It sounds like you're claiming that, in Euclidean geometry, 'distance' is a coordinate-dependent notion. If that really is what you're saying, then I posit that you need to review elementary geometry before continuing to reflect upon physics.

 

[Hurkyl]

Can a geodesic in GR be a straight line?

All geodesics are, by definition, straight lines.

What curves a geodesic in GR?

Geodesics are not curved.



The geometry of space-time yields a meaning to the term "straight" (which I will henceforth call 'intrinsic-straight'). Among other things, the path traced out by a geodesic is defined to be 'intrinsic-straight'.

There is a 'usual' way to attach an affine structure to R^4, and this gives another meaning to the term "straight" (which I wlil henceforth call 'coordinate-straight'). There is no good reason to do so -- it's simply that in a different context, we often use the same set R^4 as the underlying set of an affine space, so there is a temptation to invoke that affine structure in this context.

The point is, when you select coordinates, that affine structure usually has absolutely nothing to do with the geometry of space-time; the notions of intrinsic-straightness and coordinate-straightness are different. An intrinsic-straight path will generally be coordinate-curved. Similarly, a coordinate-straight path will generally be intrinsic-curved.

 

[Antenna Guy]

The point is, when you select coordinates, that affine structure usually has absolutely nothing to do with the geometry of space-time; the notions of intrinsic-straightness and coordinate-straightness are different. An intrinsic-straight path will generally be coordinate-curved. Similarly, a coordinate-straight path will generally be intrinsic-curved.

I think I follow.

Would it be correct to say that a particle following a geodesic (intrinsic-straight path in space-time) could be mapped as a classical trajectory (coordinate-curved path in space)?

Regards,

Bill

 

[Dale]

Hi bernhard, I wondered how long it would take for you to join the conversation!

Please follow the following thoughts:
1.Einstein's transformation equations hold only with Einstein synchronized clocks t(E) and t"(E).

Yes, everyone already agreed with that. The underlying Minkowski geometry is present regardless of the synchronization convention.

 

[bernhard.rothenstein]

convention simultaneity

Hi bernhard, I wondered how long it would take for you to join the conversation! Yes, everyone already agreed with that. The underlying Minkowski geometry is present regardless of the synchronization convention.

what about my thoughts 2 and 3?

 

[Dale]

what about my thoughts 2 and 3?

For 3 the ether is non-physical and the absolute frame is arbitrary, so I don't care.

For 2 I don't know enough about the specific transformations you referenced to comment about them in particular. But I already gave general comments about coordinate systems and synchronization conventions in https://www.physicsforums.com/showpost.php?p=1720437&postcount=26".

 

[Ken G]

If you are going to try to win an argument by appeal to authority you should at least try to do better than Wikipedia. RL Faber. "Differential Geometry and Relativity Theory: An Introduction" has a whole chapter entitled "Special Relativity: the Geometry of Flat Spacetime". Or MS Parvez. "On the theory of flat spacetime" which says in the abstract "Special relativity, in essence, is a theory of four-dimensional flat spacetime".

Both of those chapter headings are of course true (I never said otherwise), and neither are responsive to the issue of "what is the word 'special' there to imply".

I note that you didn't address the point I made about geodesics.

It required no comment, I am aware that geodesics in general relativity become straight lines in special relativity. Again, it's simply not responsive to the question of the meaning of "special", and again I repeat that this comes from the specialness of the treatment of inertial frames. That is the key element that distinguishes the approach of general and special relativity, in any kind of spacetime.

Still, this is not a terribly important semantic question-- you are welcome to your opinion on that matter, and it may be unanswerable because both the name and the theory have evolved so much that multiple interpretations of that process may be possible.

Sorry Ken, you need to read up a little more. The first postulate is in fact closer to your second statement than the first. The first postulate is, in Einstein's words, "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good ... The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion".

Again, I can't agree that those statements support your contention. When Einstein said "all frames for which the equations hold good", he obviously means "all inertial frames". And in the second sentence, he said that the changes have to be referred to the coordinates used by inertial observers, but if one is going to "refer" to coordinates willy nilly, there is no problem with simply using "the coordinates of the King", and be done. His point is that you can choose any inertial frame, i.e., any of the special frames, to refer to, and use that special frame, where the postulates apply, to translate between measurements by noninertial observers. It is implicit that the coordinates of that inertial frame correspond to the measurements of a hypothetical inertial observer, i.e. of the special class of observers that define special relativity. Whether or not that special observer actually exists is irrelevent, physics uses hypothetical observers all the time, as did Einstein.

 

[Dale]

Ken, this conversation is getting repetitive and boring.

In summary:
1) The general theory of relativity simplifies to the special theory of relativity in flat spacetime, hence SR is a special case of GR.
2) The first postulate refers to inertial reference frames, not inertial observers.
3) Einstein synchronization is a convention.
4) The fundamental and coordinate independent concept of SR is the Minkowski geometry of spacetime.

I'm done.

 

[Ken G]

Again, you're confusing "the metric" with "coordinate representation of the metric"

Normally, you start out by defining a bunch of vectors, and a metric is a way to take those vectors two at a time and associate a number with each pair in a bilinear way. Unless you plan to enumerate every such pairing, you will need a convenient way to name the vectors, such that the metric can work automatically on that naming convention. That's called a coordinatization. Then you name the metric by how it functions on that vector-naming convention, that's what is meant by the "Minkowski metric". Thus, the naming convention on the vectors is presumed in the naming of the metric. It is not I who confuses that with what a metric is-- it is the way the Minkowski metric is taught that does that, and this is very much the point of the thread.

It sounds like you're claiming that, in Euclidean geometry, 'distance' is a coordinate-dependent notion. If that really is what you're saying, then I posit that you need to review elementary geometry before continuing to reflect upon physics.

I posit you need to read my words more carefully. What I was saying is that the way we produce a concept of distance is by the use of a metric, or an inner product if you will. You said that Euclidean geometry "preserves" lengths and angles. I presumed your use of the word "preserve" meant "leaves invariant under some type of mapping of the space into itself". Then I pointed out that, if you did mean that, the statement only holds on the subclass of mappings that are "orthonormal" under the action of your metric. If you only meant that distances are by definition the same no matter how you coordinatize the space, then (1) there's no meaning to the word "preserve", as there's nothing to preserve, you've already declared by fiat what the distance is, (2) that would hold in any geometry that admits a metric, and (3) that is what I meant by "redefining what you mean by distance" (I probably should have said redefine what you mean by an inner product that magically knows what the vector was before its name got changed).

Oh, and drop the haughtiness, it's not being backed up.

 

[Ken G]

1) The general theory of relativity simplifies to the special theory of relativity in flat spacetime, hence SR is a special case of GR.

Agreed, nor did I ever express any disagreement. The reason behind this is summed up well by John Baez(http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html):
"The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is treated as absolute. In general relativity all motion is relative. To accommodate this change general relativity has to use curved space-time. In special relativity space-time is always flat."

So there it is, we see that indeed special relativity does involve the restriction to flat spacetime, and the reason for this, and where the word "special" comes from, is that inertial frames are treated differently than noninertial frames, i.e., inertial frames are special in that theory. Take it up with Dr. Baez, I can't delve any deeper than I have already.

2) The first postulate refers to inertial reference frames, not inertial observers.

Indeed, but be careful you are not implying that the explicit reference to inertial frames does not include the all-important implicit reference to inertial (possibly hypothetical) observers. How else do you plan to define an inertial frame but by using observers, possibly hypothetical, with accelerometers that read zero? (See the definition you linked-- unfortunately that same library has no entry for "inertial reference frame".)

3) Einstein synchronization is a convention.

Agreed, when we have access to the concept of the measurements of hypothetical inertial observers. Then when we add the Einstein convention, we get that the connection between those observer coordinates is the Lorentz transformation, and we find that the invariant distance is given by the Minkowski metric. If we don't use the Einstein convention, we get neither the Lorentz transformation nor invariance of the Minkowski metric as it is normally expressed (or alternatively, we need to define a new Minkowski metric commensurate with the new time coordinatization).

4) The fundamental and coordinate independent concept of SR is the Minkowski geometry of spacetime.

If that were true, there'd be no need for this thread. But it isn't. "Minkowski geometry", as it is generally used, means a geometry spawned by an inner product that deviates from Euclidean by a -1 in one of the terms, using a particular choice of basis vectors chosen from a special class that represent a particular physically-motivated ordering of events by inertial observers. That is not "coordinate independent", because that particular metric, defined in the Minkowski way, is only invariant when acting on those special coordinates generated by inertial observers, coordinates which are connected by Lorentz transformations, which are of course the orthonormal transformations under the action of the Minkowksi metric (which is why it is invariant for those transformations). This I would say is precisely the basis of Baez's remark: "In special relativity accelerating frames are different from inertial frames."

Put mathematically, it is a very basic theorem of metric spaces that g(x,y)=g(Lx,Ly) only works if L is an orthonormal transformation (indeed, that defines the orthonormal transformations under the action of g). Ergo "Minkowski geometry" is very much the geometry associated with the Lorentz transformations and the Einstein simultaneity convention. These form a subclass of linear coordinatizations, and hence "Minkowski geometry" is not "coordinate independent". Just what is coordinate independent in all this is precisely the question behind this thread.

I'm done.

We all decide how much we want to know.

 

[Ken G]

1.Einstein's transformation equations hold only with Einstein synchronized clocks t(E) and t"(E).

Yes, the coordinate form of the standard "Lorentz transformation" between inertial frames (as defined by a population of hypothetical inertial observers) requires that inertial observers use the Einstein simultaneity convention.

2.the t(E) and t'(E) readings could be brought in a physically correct relationship with the readings of other clocks synchonized in a different way.

Yes, I would say that any arbitrary prescription could be used to synchronize clocks that merely followed some very weak constraints (supportive of metric spaces), and it would merely spawn a new way to transform between the inertial observers' coordinates. The special treatment of those observers would still allow the first postulate of SR to apply, but the second postulate would be lost. In that sense, I see the second postulate as superfluous, and Einstein's simultaneity convention should be elevated to the level of a postulate if one wanted to work in the standard coordinatization. If one wanted a coordinate-free treatment, one would simply assert that the speed of light is whatever is necessary to allow the first postulate to hold.

3.It is considered that under such conditions the reference frame I is in absolure rest relative to the ether the motion of I' relative to it having an absolute character confering to I' some properties (anisotropy).

Yes, it is a matter of sheer preference, a la Occam's Razor, to exclude anisotropy. In other words, if we later found some new physics that required anisotropy, no previous experiments would suddenly seem strange, we would merely have to use a different simultaneity convention and/or a different status of what are the "special" reference frames.

4.My oppinion is that those properties are merely introduced by the shift from the reading t(E) to the reading t(v). Using different physically correct relationships between t(E) and the reading t of a differently synchronized clock we obtain different transformation equations which confer different properties to I'.

To me, the key unanswered issue is, "how should we think of all this so that none of the arbitrary choices matter, i.e., what possibilities are ruled out by experiment and what is just what we accept from our preference for simplicity?"